3.552 \(\int \frac{12-x}{4+x+\sqrt{-9+6 x}} \, dx\)

Optimal. Leaf size=71 \[ -x+2 \sqrt{3} \sqrt{2 x-3}+10 \log \left (x+\sqrt{3} \sqrt{2 x-3}+4\right )-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]

[Out]

-x + 2*Sqrt[3]*Sqrt[-3 + 2*x] - 21*Sqrt[3/2]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqrt
[6])] + 10*Log[4 + x + Sqrt[3]*Sqrt[-3 + 2*x]]

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Rubi [A]  time = 0.188021, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -x+2 \sqrt{3} \sqrt{2 x-3}+10 \log \left (x+\sqrt{3} \sqrt{2 x-3}+4\right )-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]

[Out]

-x + 2*Sqrt[3]*Sqrt[-3 + 2*x] - 21*Sqrt[3/2]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqrt
[6])] + 10*Log[4 + x + Sqrt[3]*Sqrt[-3 + 2*x]]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 \sqrt{3} \sqrt{2 x - 3} + 10 \log{\left (2 x + 2 \sqrt{3} \sqrt{2 x - 3} + 8 \right )} - \frac{21 \sqrt{6} \operatorname{atan}{\left (\sqrt{2} \left (\frac{\sqrt{2 x - 3}}{4} + \frac{\sqrt{3}}{4}\right ) \right )}}{2} - \int ^{\sqrt{2 x - 3}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((12-x)/(4+x+(-9+6*x)**(1/2)),x)

[Out]

2*sqrt(3)*sqrt(2*x - 3) + 10*log(2*x + 2*sqrt(3)*sqrt(2*x - 3) + 8) - 21*sqrt(6)
*atan(sqrt(2)*(sqrt(2*x - 3)/4 + sqrt(3)/4))/2 - Integral(x, (x, sqrt(2*x - 3)))

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Mathematica [A]  time = 0.0452376, size = 70, normalized size = 0.99 \[ \frac{1}{6} (9-6 x)+2 \sqrt{6 x-9}+10 \log \left (6 x+6 \sqrt{6 x-9}+24\right )-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]

[Out]

(9 - 6*x)/6 + 2*Sqrt[-9 + 6*x] - 21*Sqrt[3/2]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqr
t[6])] + 10*Log[24 + 6*x + 6*Sqrt[-9 + 6*x]]

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Maple [A]  time = 0.006, size = 54, normalized size = 0.8 \[ 2\,\sqrt{-9+6\,x}+{\frac{3}{2}}-x+10\,\ln \left ( 24+6\,x+6\,\sqrt{-9+6\,x} \right ) -{\frac{21\,\sqrt{6}}{2}\arctan \left ({\frac{\sqrt{6}}{24} \left ( 6+2\,\sqrt{-9+6\,x} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((12-x)/(4+x+(-9+6*x)^(1/2)),x)

[Out]

2*(-9+6*x)^(1/2)+3/2-x+10*ln(24+6*x+6*(-9+6*x)^(1/2))-21/2*6^(1/2)*arctan(1/24*(
6+2*(-9+6*x)^(1/2))*6^(1/2))

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Maxima [A]  time = 0.801669, size = 69, normalized size = 0.97 \[ -\frac{21}{2} \, \sqrt{6} \arctan \left (\frac{1}{12} \, \sqrt{6}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt{6 \, x - 9} + 10 \, \log \left (6 \, x + 6 \, \sqrt{6 \, x - 9} + 24\right ) + \frac{3}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 12)/(x + sqrt(6*x - 9) + 4),x, algorithm="maxima")

[Out]

-21/2*sqrt(6)*arctan(1/12*sqrt(6)*(sqrt(6*x - 9) + 3)) - x + 2*sqrt(6*x - 9) + 1
0*log(6*x + 6*sqrt(6*x - 9) + 24) + 3/2

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Fricas [A]  time = 0.266952, size = 90, normalized size = 1.27 \[ -\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} x + 21 \, \sqrt{3} \arctan \left (\frac{1}{12} \, \sqrt{3}{\left (\sqrt{2} \sqrt{6 \, x - 9} + 3 \, \sqrt{2}\right )}\right ) - 10 \, \sqrt{2} \log \left (x + \sqrt{6 \, x - 9} + 4\right ) - 2 \, \sqrt{2} \sqrt{6 \, x - 9}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 12)/(x + sqrt(6*x - 9) + 4),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*(sqrt(2)*x + 21*sqrt(3)*arctan(1/12*sqrt(3)*(sqrt(2)*sqrt(6*x - 9)
+ 3*sqrt(2))) - 10*sqrt(2)*log(x + sqrt(6*x - 9) + 4) - 2*sqrt(2)*sqrt(6*x - 9))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x + \sqrt{3} \sqrt{2 x - 3} + 4}\, dx - \int \left (- \frac{12}{x + \sqrt{3} \sqrt{2 x - 3} + 4}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((12-x)/(4+x+(-9+6*x)**(1/2)),x)

[Out]

-Integral(x/(x + sqrt(3)*sqrt(2*x - 3) + 4), x) - Integral(-12/(x + sqrt(3)*sqrt
(2*x - 3) + 4), x)

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GIAC/XCAS [A]  time = 0.269275, size = 117, normalized size = 1.65 \[ -\frac{1}{6} \, \sqrt{3} \sqrt{2}{\left (10 \, \sqrt{3} \sqrt{2}{\rm ln}\left (33\right ) - 63 \, \arctan \left (\frac{1}{4} \, \sqrt{3} \sqrt{2}\right )\right )} - \frac{21}{2} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{12} \, \sqrt{3} \sqrt{2}{\left (\sqrt{6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt{6 \, x - 9} + 10 \,{\rm ln}\left (6 \, x + 6 \, \sqrt{6 \, x - 9} + 24\right ) + \frac{3}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 12)/(x + sqrt(6*x - 9) + 4),x, algorithm="giac")

[Out]

-1/6*sqrt(3)*sqrt(2)*(10*sqrt(3)*sqrt(2)*ln(33) - 63*arctan(1/4*sqrt(3)*sqrt(2))
) - 21/2*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)*(sqrt(6*x - 9) + 3)) - x +
2*sqrt(6*x - 9) + 10*ln(6*x + 6*sqrt(6*x - 9) + 24) + 3/2